category theory applications

0.1 Category Theory Applications in Physics

A ‘categorification’ of theoretical physics (including quantum field theories) began as early as 1968 [5, 7, 8]; this ‘categorification’ process in physics continues today, especially after 1985 ([15] and references cited therein). Other recent applications are in computer programming theory, automata theory, ontology/categorical ontology [18] and linguistics.

0.2 Category Theory Applications in Mathematical Biophysics

categories of sets were first introduced in mathematical biophysics in 1958 [31, 32], followed by the introduction of biotheoretical models in categories with structure in 1968-1971 [5, 6, 7, 8]. For a review of the earlier work see ref. [15].

References

[1] Baez, J. & Dolan, J., 1998a, Higher-Dimensional Algebra III. n-Categories and the Algebra of Opetopes, in: Advances in Mathematics, 135, 145–206.

[2] Baez, J. & Dolan, J., 1998b, “Categorification”, Higher Category Theory, Contemporary Mathematics, 230, Providence: AMS, 1–36.

[3] Baez, J. & Dolan, J., 2001, From Finite Sets to Feynman Diagrams, in Mathematics Unlimited – 2001 and Beyond, Berlin: Springer, 29–50.

[4] Baez, J., 1997, An Introduction to n-Categories, in Category Theory and Computer Science, Lecture Notes in Computer Science, 1290, Berlin: Springer-Verlag, 1–33.

[5] Baianu, I. and M. Marinescu: 1968, Organismic Supercategories: Towards a Unitary Theory of Systems. Bulletin of Mathematical Biophysics 30, 148-159.

[6] Baianu, I.C.: 1970, Organismic Supercategories: II. On Multistable Systems. Bulletin of Mathematical Biophysics, 32: 539-561.

[7] Baianu,I.C. : 1971a, Organismic Supercategories and Qualitative Dynamics of Systems. Ibid., 33 (3), 339–354.

[8] I.C. Baianu: 1971b, Categories, Functors and Quantum Algebraic Computations, in P. Suppes (ed.), Proceed. Fourth Intl. Congress Logic-Mathematics-Philosophy of Science, September 1–4, 1971, University of Bucharest.

[9] I.C. Baianu: Łukasiewicz-Topos Models of Neural Networks, Cell Genome and Interactome Nonlinear Dynamics). CERN Preprint EXT-2004-059. Health Physics and Radiation Effects (June 29, 2004).

[10] I.C. Baianu and D. Scripcariu: 1973, On Adjoint Dynamical Systems. The Bulletin of Mathematical Biophysics, 35(4), 475–486.

[11] I.C. Baianu: 1973, Some Algebraic Properties of (M,R) – Systems. Bulletin of Mathematical Biophysics 35, 213-217.

[12] I.C. Baianu and M. Marinescu: 1974, A Functorial Construction of (M,R)– Systems. Revue Roumaine de Mathematiques Pures et Appliquees 19: 388-391.

[13] I.C. Baianu: 1977, A Logical Model of Genetic Activities in Łukasiewicz Algebras: The Non-linear Theory. Bulletin of Mathematical Biophysics, 39: 249-258.

[14] I.C. Baianu: 1980, Natural Transformations of Organismic Structures. Bulletin of Mathematical Biophysics 42: 431-446.

[15] Baianu, I.C.: 1987. Mathematical Models in Medicine, vol. 7., Ch.11 Pergamon Press, New York, 1513 -1577; URLs: CERN Preprint No. EXT-2004-072:, available here as PDF, or as as an archived html document.

[16] Baianu, I. C., Glazebrook, J. F. and G. Georgescu: 2004, Categories of Quantum Automata and N-Valued Łukasiewicz Algebras in Relation to Dynamic Bionetworks, (M,R)–Systems and Their Higher Dimensional Algebra, PDF’s of Abstract and Preprint of Report.

[17] Baianu, I.C. Brown R., J. F. Glazebrook, and Georgescu G.: 2006, Complex Nonlinear Biodynamics in Categories, Higher Dimensional Algebra and Łukasiewicz–Moisil Topos: Transformations of Neuronal, Genetic and Neoplastic networks, Axiomathes 16 Nos. 1–2, 65–122.

[18] Baianu, I.C., R. Brown and J. F. Glazebrook: 2007b, A Non-Abelian, Categorical Ontology of Spacetimes and Quantum Gravity, Axiomathes, 17: 169-225.

[19] Eilenberg, S. and S. Mac Lane.: 1942, Natural Isomorphisms in Group Theory., American Mathematical Society 43: 757-831.

[20] Eilenberg, S. and S. Mac Lane: 1945, The General Theory of Natural Equivalences, Transactions of the American Mathematical Society 58: 231-294.

[21] Eilenberg, S. & Cartan, H., 1956, Homological Algebra, Princeton: Princeton University Press.

[22] Grothendieck, A. et al., Séminaire de Géométrie Algébrique, Vol. 1–7, Berlin: Springer-Verlag.

[23] Grothendieck, A., 1957, ”Sur Quelques Points d’algèbre homologique”, Tohoku Mathematics Journal, 9, 119–221.

[24] Lawvere, F. W., 1964, ”An Elementary Theory of the Category of Sets”, Proceedings of the National Academy of Sciences U.S.A., 52, 1506–1511.

[25] Lawvere, F. W., 1965, ”Algebraic Theories, Algebraic Categories, and Algebraic Functors”, Theory of Models, Amsterdam: North Holland, 413–418.

[26] Lawvere, F. W., 1966, ”The Category of Categories as a Foundation for Mathematics”, Proceedings of the Conference on Categorical Algebra, La Jolla, New York: Springer-Verlag, 1–21.

[27] MacLane, S., 1997, Categories for the Working Mathematician, 2nd edition, New York: Springer-Verlag.

[28] Eilenberg, S. and S. Mac Lane.: 1942, Natural Isomorphisms in Group Theory., American Mathematical Society 43: 757-831.

[29] Eilenberg, S. and S. Mac Lane: 1945, The General Theory of Natural Equivalences, Transactions of the American Mathematical Society 58: 231-294.

[30] Baianu I. C., Brown R., Georgescu G. and J. F. Glazebrook: 2006b, Complex Nonlinear Biodynamics in Categories, Higher Dimensional Algebra and Łukasiewicz–Moisil Topos: Transformations of Neuronal, Genetic and Neoplastic Networks., Axiomathes, 16 Nos. 1–2: 65–122.

[31] Rosen, R.: 1958a, A Relational Theory of Biological Systems., Bulletin of Mathematical Biophysics 20: 245-260.

[32] Rosen, R.: 1958b, The Representation of Biological Systems from the Standpoint of the Theory of Categories., Bulletin of Mathematical Biophysics 20: 317-341.

[33] See also a more extensive category theory bibliography

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